Earlier today I set you these two puzzles. Here they are again with answers.
1. Adam’s apples
Adam buys a number of apples in one shop at the rate of 3 for £1.
He then buys the same number of apples in a different shop at a rate of 5 for £1.
What was the average number of apples bought for each pound spent?
Note: please don’t fall into the trap.
Solution
This is one of the problems when you are deliberately led to give the wrong answer. At first, one might think the average number of apples is 4 for a £1, but that is incorrect in the scenario described.
Imagine Adam buys 15 apples in each shop. He would spend £5 in shop 1, and £3 in shop 2. In total he has spent £8 and has 30 apples. For each pound spent, he has thus bought 30/8, or 3.75 apples.
If Adam had spent an equal number of pounds per shop, then the average number of apples per pound spent would be 4. But the question makes it clear that what is equal is the number of apples bought, not the money spent.
2. I can’t get no satisfaction
1 + x + x2 = y2
Show that there are no positive whole numbers x and y that satisfy the above equation.
Solution
You prove this by contradiction. You also need to remember how to multiply terms in brackets.
If x is a positive whole number, then x2 < (x + 1)2
The term (x + 1)2 is equal to (x + 1)(x + 1), which expands out to x2 + 2x + 1.
Since x is a positive whole number, we also know that
x2 < x2 + x + 1 < x2 + 2x + 1
From the equation in the question
x2 < y2 < x2 + 2x + 1
x2 < y2 < (x + 1)2
In other words the square of y must be in between the squares of x and x + 1. So y must be a positive whole number between x and x + 1, which is impossible. QED.
Thanks to Owen O’Shea, author of The Call of Coincidence: Mathematical Gems, Peculiar Patterns and more Stories of Numerical Serendipity, the source of today’s puzzles.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
